How to make a soft body stay static and not vacillate at all?
First method:

Simplest solution would be to reduce Cube mass to something like 0.5. This will prevent wobbling but it will keep Cube in place so Sphere will go trough it.



Also you will have limited control over Cube behavior.

Second method:

This is what I can recommend. In this method we need to first settle down Cube on Collision Plane, and then affect it by Sphere. 

In this method you have full control over collisions. In my example I've used settings that will allow to move Cube by Sphere, but you can also change it to destroy Cube without moving it or just "melt" into it.



First of all you will need simple Plane with Collision turned on and Soft Body and Cloth > Outer > 0.001

Then set Cube Soft Body values as below:



Pull - lower value, lower impact over whole Cube.

Bending - higher value will move Cube, lower will keep it in place.

Test Blend file: